# 10th Maths Paper Solutions Set 1 : CBSE Delhi Previous Year 2014

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.
(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.
(iv) Use of calculators is not permitted.
Q1 :

The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:
(A) –3
(B) 4
(C) 5
(D) 2

The first three terms of an AP are 3y1, 3y+5 and 5y+1, respectively.

We need to find the value of y.
We know that if a, b and c are in AP, then:
b − a = c − b ⇒ 2b = a + c

2(3y+5) = 3y  1 + 5y + 16y + 10 = 8y10 = 8y 6y 2y = 10y = 5

Hence, the correct option is C.

Q2 :

In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is : (A) 3.8
(B) 7.6
(C) 5.7
(D) 1.9 It is known that the length of the tangents drawn from an external point to a circle are  equal.

∴ QP = PT = 3.8 cm               ...(1)

PR = PT = 3.8 cm                 ...(2)

From equations (1) and (2), we get:

QP = PR = 3.8 cm

Now, QR = QP + PR
= 3.8 cm + 3.8 cm
= 7.6 cm

Hence, the correct option is B.

Q3 :

In Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals: (A) 67°
(B) 134°
(C) 44°
(D) 46°

Given: ∠QPR = 46°
PQ and PR are tangents.
Therefore, the radius drawn to these tangents will be perpendicular to the tangents.
So, we have OQ ⊥ PQ and OR ⊥ RP.
⇒ ∠OQP = ∠ORP = 90
So, in quadrilateral PQOR, we have
∠OQP +∠QPR + ∠PRO + ∠ROQ = 360
⇒ 90° + 46° + 90° + ∠ROQ = 360
⇒ ∠ROQ = 360 − 226 = 134

Hence, the correct option is B.

Q4 :

A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:
(A) 43

(B) 43

(C) 22

(D) 4

Q5 :

If two different dice are rolled together, the probability of getting an even number on both dice, is:
(A) 136
(B) 12
(C) 16
(D) 14

Q6 :

A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:
(A) 23

(B) 16

(C) 13

(D) 1130

Q7 :

If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:
(A) −63
(B) 63
(C) 60
(D) −60

Q8 :

The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:
(A) 3
(B) 5
(C) 4
(D) 6

Q9 :

Solve the quadratic equation 2x2 + axa2 = 0 for x.

Q10 :

The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

Q11 :

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

Q12 :

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.

Q13 :

Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

Q14 :

In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14) Q15 :

Solve the equation 4x3=52x+3; x0, 32, for x.

Q16 :

If the seventh term of an AP is 19 and its ninth term is 17, find its 63rd term.

Q17 :

Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

Q18 :

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also find the length of AB.

Q19 :

Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [Use 3=1.73]

Q20 :

If the points A(−2, 1), B(a, b) and C(4, −1) are collinear and ab = 1, find the values of a and b.

Q21 :

In  Fig 4, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region. [Use π = 3.14 and 3=1.73] Q22 :

In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π = 3.14] Q23 :

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

Q24 :

A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 112 cm, find the length of the wire.

Q25 :

The difference of two natural numbers is 5 and the difference of their reciprocals is 110. Find the numbers.

Q26 :

Prove that the length of the tangents drawn from an external point to a circle are equal.

Q27 :

The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

Q28 :

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is:
(i) an odd number
(ii) a multiple of 5
(iii) a perfect square
(iv) an even prime number

Q29 :

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, – 3). Also find the value of x.

Q30 :

Find the values of k for which the quadratic equation (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. Also find these roots.

Q31 :

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

Q32 :

Prove that a parallelogram circumscribing a circle is a rhombus.

Q33 :

Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 25th of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?